theorem LMThMBF3:
  for V being finite-rank free Z_Module,
  b1, b2 being OrdBasis of V holds
  AutMt(id(V), b1, b2) is Matrix of rank V,INT.Ring
  proof
    let V be finite-rank free Z_Module,
    b1, b2 be OrdBasis of V;
    set n = rank V;
    A1: len b1 = rank V by ThRank1;
    A2: len b2 = rank V by ThRank1;
    P0: len AutMt(id(V),b1,b2) = len b1 by Def8;
    per cases;
    suppose X1: len b1 = 0;
      then len AutMt(id(V), b1, b2) = 0 by Def8;
      then AutMt(id(V), b1, b2) = {};
      hence thesis by A1,X1,MATRIX_0:13;
    end;
    suppose P1: 0 < len b1; then
      width AutMt(id(V),b1,b2) = len b2 by Th39;
      hence thesis by P0,P1,A1,A2,MATRIX_0:20;
    end;
  end;
